Metamath Proof Explorer

Theorem dalem15

Description: Lemma for dath . The axis of perspectivity X is a line. (Contributed by NM, 21-Jul-2012)

Ref Expression
Hypotheses dalema.ph 
|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalemc.l 
|- .<_ = ( le ` K )
dalemc.j 
|- .\/ = ( join ` K )
dalemc.a 
|- A = ( Atoms ` K )
dalem15.m 
|- ./\ = ( meet ` K )
dalem15.n 
|- N = ( LLines ` K )
dalem15.o 
|- O = ( LPlanes ` K )
dalem15.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem15.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem15.x 
|- X = ( Y ./\ Z )
Assertion dalem15 
|- ( ( ph /\ Y =/= Z ) -> X e. N )

Proof

Step Hyp Ref Expression
1 dalema.ph 
 |-  ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
2 dalemc.l 
 |-  .<_ = ( le ` K )
3 dalemc.j 
 |-  .\/ = ( join ` K )
4 dalemc.a 
 |-  A = ( Atoms ` K )
5 dalem15.m 
 |-  ./\ = ( meet ` K )
6 dalem15.n 
 |-  N = ( LLines ` K )
7 dalem15.o 
 |-  O = ( LPlanes ` K )
8 dalem15.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem15.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem15.x 
 |-  X = ( Y ./\ Z )
11 eqid 
 |-  ( LVols ` K ) = ( LVols ` K )
12 eqid 
 |-  ( Y .\/ C ) = ( Y .\/ C )
13 1 2 3 4 7 11 8 9 12 dalem14 
 |-  ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) e. ( LVols ` K ) )
14 1 dalemkehl 
 |-  ( ph -> K e. HL )
15 1 dalemyeo 
 |-  ( ph -> Y e. O )
16 1 dalemzeo 
 |-  ( ph -> Z e. O )
17 3 5 6 7 11 2lplnmj 
 |-  ( ( K e. HL /\ Y e. O /\ Z e. O ) -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) )
18 14 15 16 17 syl3anc 
 |-  ( ph -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) )
19 18 adantr 
 |-  ( ( ph /\ Y =/= Z ) -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) )
20 13 19 mpbird 
 |-  ( ( ph /\ Y =/= Z ) -> ( Y ./\ Z ) e. N )
21 10 20 eqeltrid 
 |-  ( ( ph /\ Y =/= Z ) -> X e. N )